The second Particles System not working Properly
There should be 2 particle systems on the object (there's only one in your file). The second particle system modifier should be in the bottom of the stack.



You will setup it as desired, the main option to change would be start and end frame.
Starting frame should be roughly the same as the end frame of the first system. Values from the lesson didn't work well so I changed them - while particles of first system die at frame 26, particles of the second one burn at 23.  

However to make second particle system emit particles from the faces exploded by the Explode modifier you should enable Use Modifier Stack option:



The reason this wasn't mentioned in tutorial is because Use Modifier Stack option was added starting from version 2.68.